Here is a "picture" of what you have described in words, wlm2.
Code:
[Router with DHCP server]--{wireless}--[Laptop]--{wired to vr0}--[ALIX6E1 with DHCP client]
If this is correct, you have a DHCP server at one end of your network, and a DHCP client at the other end of the network. They are not communicating,
because they are on different physical networks.
DHCP requires the client and server to be on the same physical network, or, to have the traffic proxied through a router.
Windows does not default to acting as a router, but it can be set to be one, through a Microsoft technology they call "Internet Connection Sharing", or ICS. That has limited capabilities, acting as a NAT router only and providing limited DHCP services to the subnet under its authority. It cannot proxy DHCP services from your ISP's router, nor, with the topology it can provide, should it, since the "wired" network would be a separate TCP/IP subnet that has its addresses translated by Windows.